Persistence in food webs—I Lotka-Volterra food chains

Bulletin of Mathematical Biology, Vol. 41, pp. 877 891 Pergamon Press Ltd. 1979. Printed ill Great Britain © Society for Mathematical Biology

0007-4985/79/1201-0877 $02.0O/0

PERSISTENCE IN F O O D W E B S - - I L O T K A - V O L T E R R A F O O D CHAINS

ITHOMAS C. GARD Department of Mathematics, University of Georgia, Athens, Georgia 30602, U.S.A. ITHOMAS G. HALLAM Departments of Mathematics and Ecology, University of Tennessee, Knoxville, Tennessee 37916, U.S.A.

Persistenc~extinction in simple food chains modelled by Lotka Volterra dynamics is governed by a single parameter which depends upon the interspecific interaction coefficients, the intraspecific interaction coefficients, and the length o f the food chain. In persistent systems with nonzero carrying capacity, two new features predominate. Trophic level influence factors relate persistence on different trophic levels and determine, in conjunction with the persistence parameter , the magnitude o f persistence. Equilibrium component ordering, which results in persistent systems, mandates once again that systems need to be studied on the complete ecosystem level; static field measurements reflect species location in the food chain, the total length o f the food chain and assume characteristics according to these factors.

I. Introduction. Of numerous ecosystem stability considerations, probably none .affects ecosystem structure as drastically as the extinction of a biological species (e.g. Paine, 1966). Public concern about projects such as the TVA Tellico Dam Project, where the persistence of the snail darter Percina tanasi is in doubt (Holden, 1977) demonstrates that extinction of a biotic component of a food chain is topical as well as ecologically fundamental. In view of the importance of persistence-extinction phenomena, it is somewhat surprising that the modelling and analysis of ecosystems have not concentrated foremost upon species survival. The influence of stability developments in the engineering and mathematical communities was, undoubtedly, initially too inviting and Overwhelming. 877

878

T H O M A S C. GARD AND T H O M A S G. HALLAM

Ecosystem stability considerations are maturing and evolving with practicality and realism in focus (Holling, 1973; May, 1974; Maynard Smith, 1974; Patten, 1974); however, a generally applicable theory of stability for ecosystems is, at best, in infancy. Goodman (1975) states that "minimally, stability means persistence." In this article, we initiate a study of the fundamental aspect of ecosystem stability, persistence extinction, with emphasis upon developing techniques which might be applicable in a general setting. Persistence attributes of simple deterministic food chains of arbitrary length which are modelled by Lotka-Volterra dynamics are investigated. Here a simple food chain has trophic levels which are functionMly regarded as a single species. The dynamics of each trophic level is governed by the adjoining trophic levels which immediately precede or succeed it. Persistence is, in general, a global property of a dynamical system; it is not dependent upon interior solution space structure but is dependent upon solution behavior near extinction boundaries. Equilibrium analysis used in conjunction with linearization techniques has been a principal tool used for studying the survival problem (Freedman and Waltman, 1977; May, 1974; Rescigno and Jones, 1972; Saunders and Bazin, 1975). Such approaches have not been particularly fruitful for higher dimensional food chains basically due to intrinsic complex analytical and topological problems; for example, classification of recurrent solutions of higher dimensional dynamical systems is, at best, difficult. Other techniques, for example, graph theoretical ones, have been of some utility in higher dimensional food chains (Yorke and Anderson, 1973). The qualitative approach employed here introduces the concept of a persistence or extinction function, which is essentially an appropriate system transformation in the Liapunov tradition. This technique yields (global) persistence or extinction results from the system structure without a priori information of the asymptotic character of the model solutions.

2. Preliminary Terminology and Techniques. Let R+ denote the nonnegative real numbers and R~ the non-negative cone in R"; that is, R'$ ={xeRn:x=(xl,...,x,,) where xieR+, i=1, 2 .... ,n}. The positive cone in R n is R~={x~R"+:x=(xl,...,xn), xi>0}. The models considered here are of the form

x~=xifi(xl,x2 ..... x,),

i = 1 , 2 .... ,n;'=d/dt;

(1)

where each ./~ is a continuous function from R'$ to R and is sufficiently smooth to guarantee that initial value problems for (1), with the initial

P E R S I S T E N C E I N F O O D WEBS~ i. L O T K A V O L T E R R A F O O D C H A IN S

879

position having non-negative components, have unique solutions. The primary theme develops persistence of a food chain modelled by (1) in the following sense.

Definition 1. System (1) is persistent if each solution q~= q~(t) of (1) with q~(0)~R~ satisfies lim sup~_,~q~i(t)>0 for all i, l < i < n , and re(0, Tq~] where [0, T4o) is the maximal interval of existence of q~. If (1) is not persistent, then there is a solution ~b with 4)(O)ER"+° where some component, say q~j, satisfies lim,~¢bi(t)=0 for some ~c in (0, T4,]. As we shall be concerned with persistence of the complete food chain, any solution o f (1) considered subsequently will tacitly have initial value in R~. I f it is known that the maximal interval of existence of all solutions of (1) is [0, oo), (that is, T~= oo) then it follows from the assumed uniqueness of solutions o f initial value problems that persistence o f (1) is determined by using only z = oo in Definition 1. In the sequel, the well known comparison technique is employed. It is convenient to categorize the differential equations employed as comparison equations in the following terms. Let co be a continuous function from R+ into R. The differential equation u'=co(u)

(2)

is of persistent type provided any solution O = 0 ( t ) of (2) with 0 ( 0 ) > 0 satisfies lim supt_~O(t ) > 0 for all z ~(0, oo]. The equation (2) is of extinction type provided any solution 0 of (2) with ~ ( 0 ) = 0 satisfies l i m , ~ p ( t ) = 0 for some -c in (0, oo]. For example, if the right side of (2) is co(u)=c~u, then (2) is of persistent type whenever a > 0 and of extinction type if a < 0. The concept o f a persistence (extinction) function is now introduced. It is tacitly required that the functions p and e used subsequently be continuous functions from R~ to R + which are continuously differentiable on R'+°.

Definition 2. A function p i s called a persistence function for system (1) if the following are satisfied: (i) p(xl,x2,...,x,,)~O if xi--*0 for some i, i = 1 , 2,...,n; (ii) p satisfies the differential inequality ~ > co(p) wherein

~(xl, x2,..., x,,)-=

x~,..., x,);

(3)

i= 1 (]Xi

and the associated comparison differential equation u'=co(u) is of persistent type.

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THOMAS C. GARD AND THOMAS G. HALLAM

Definition 3. A function e is an extinction junction Jot (1) if the following are satisfied: (iii) e(xi,x2,...,x,)-+O only if some xi--+O, i= 1,2,..., n; (iv) e satisfies the differential inequality g < co(e) where ~ is defined as in (3) and the associated comparison differential equation u'=co(u) is of extinction type. I f co(p)>0 (resp. co(e)<0), then - p (resp. e) is a Liapunov function in the sense of LaSalle (1976). The existence of a persistence (extinction) function for (1) implies the persistence (nonpersistence; i.e., extinction of a component species) of system (l). THEOREM t. Let p be a persistence.Junction for system (1). Then, for any solution qS= (qS1,q52.... , ~bn) of (1) with maximal interval of existence [0, T,/,), lira supt~qSi(t ) > 0 for each ~ ~ (0, To] and each i, i = 1, 2,..., n; that is, system (1) is persistent. Example 1. The function p(xl, X2)=XlX2/(1 tion for the symbiotic model

q-X~) is a persistence func-

X'1 =Xl(1--1Xl +2X2 +XlX2 --X2), X2 =X2(1 +X 1 --2X2);

here, p satisfies the equation t S = p ( 3 / 2 - p ) which is of persistent type and the condition p(x~,x2)-+O if xi--+0 follows by noting that Xl--0 and x 2 ~ o o are incompatible. The extinction analogue of Theorem 1 is our next result. THEOREM 2. Let e be an extinction Jhnction for (1). Then, Jbr any solution 4)= (qSl, qSz,.._,q~,) there exists an i , i = l , 2 , . . . , n and a z, 0 < z < o o , such that limt_+~bi(t ) = 0 ; that is, (1) is not persistent and extinction of some component species results. Proof of Theorem 1. Suppose that system (1) is not persistent; then, there exists a re(0, Te] and a j , l < j < n , such that limsup,_~qSa(t)~k(t), where ~(t) is the solution of the initial value problem u ' = co(u), ~(0)=p(qS(0)). The proof of Theorem 2 is similar to that above and is omitted.

PERSISTENCE IN FOOD WEBS--I. LOTKA VOLTERRA FOOD CHAINS

Example 2.

881

The system x ' l = X l ( l +aXl) ,

a>0

:c~=-x2(l÷bx2) ,

b>0

shows that considerations o f the maximal interval of existence in the above theorems are necessary. The function p(XI, Xz)=X1X2/(1 + a x 1 ÷bx2) satisfies t~=0 and is a persistence function. Solutions of x~ = - x 2 ( 1 ÷bx2) with positive initial conditions approach zero as t approaches infinity; however, solutions of the system exist only on a finite interval. As will be demonstrated below, the persistence o f (1) can sometimes be verified even if the persistence function does not satisfy the differential inequality throughout the entire region R~ but only a slab o f the form {(Xl,X2,....,x,)~R"+, 0 < x , < 2 and some 2>0}. The conclusion in these circumstances is the establishment o f a component hierarchy structured by the dominance o r t h e species which survive the struggle for existence. To illustrate, in an elementary setting, our technique for determining species survival, we consider the quadratic model x] =Xx(a + bxa -]-ex2)

x'2 = x2(e + f x 1 ÷ gx2) where a, b, c, e, f, and g are constants (not necessarily positive). Define p(x~, x2) = x l x 2 ; then

= p[(a + e) + (b + f ) x l + (c + g)x2]. Suppose we know, a priori, that solutions are bounded. If a + e > 0 , b + f > 0 and c + g > 0 then D>(a+e)p and it follows that the system is persistent. Whenever a + e > 0 and b + f > 0 , then species x2 survives since > 0 on the slab {(Xl,Xz):XlER 1, O 0 and c + g > 0 , species x~ survives. These results are, admittedly, not sharp. They are presented only to demonstrate a technique utilized subsequently in a more elaborate setting.

3. Lotka-Volterra Food Chains. By employing the techniques developed in the previous section, we now classify persistence in a simple food chain modelled by Lotka-Volterra dynamics. The results, which depend upon the interspecific and intraspecific interaction coefficients and the length o f the predator-prey system, determine persistence-extinction up to a single parameter (bifurcation) value. The presence or absence o f a carrying capacity for the food chain resource (lowest trophic level) has an interest-

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THOMAS C. GARD AND THOMAS G. HALLAM

ing effect upon the character o f the persistence criterion. The model considered here is

xl =x (a o-a xx x~ = x 2 ( - a 2 0 + a 2 1 x l - a z 3 x 3 )

(4) x'.-l=x.-l(-a.

1,o+a. 1,,, 2x,,-2-a.-1,.x.)

X'n=X,,(-a.o+an,. ix. 1)" In (4), all a u are positive constants with the exception of.all which is nonnegative. Before proceeding with our main result for (4) with carrying capacity we note that limitations on the resource level are propagated throughout the food chain.

All solutions of (4) with positive initial conditions are bounded provided al l >0. Proof. F r o m the first equation in (4), it follows that for any solution q~,

THEOREM 3.

~b'l < 01 (a 1o - al 14)1). The comparison principle establishes the bound 4)1(t)
U(N)= j

'

a/~+ 1,k x j

1

w h e r e i n any improperly indexed product is defined to be one. (The function u is constructed on R+t l satisfying u ~ o e if any xj--,oQ and so that t h e expression for fi, the derivative o f u along the integral curves of (4), can be compared to u; the simplest way to do this is to choose a linear function u = ~ = l b ~ X ) with positive coefficients bj selected so that the xix.i terms in the fi expression vanish.) F r o m (4) it follows that

=dtu(¢(t)) n-1

k--1

n--1

+ l~ ai,i+ld?n(-a.o+a.,.-lOn-1) i--1

< -mu+bl

P E R S I S T E N C E I N F O O D WEBS--I. L O T K A V O L T E R R A F O O D C H A IN S

883

where n

1

m = min ajo, b=max[(~l(2alo-a11~l)l 1~ ak+l,k. 1
01

t~= 1

Application of the comparison principle leads to

u(t ) < u(0)exp( - mr) + b/m; this implies 4i, i = 2 .... , n, is bounded and completes the p r o o f of Theorem 3. For system (4), the next theorem classifies persistence in terms of a single parameter p, defined by

a20+ E ( H

]..L=alo--(all/a21)

j=2\

.

.

j=l

.

.

.

i=]a2i+l,2i/

i=2

azi-z'2iT~- a2j.o a2i,2i-i

]a2j+ 1,o,

where

k

~n/2, ifneven = [ ( n - 1)/2, if n odd

and

l=~ ( n / 2 ) - l ' i f n e v e n [ ( n - 1)/2, if n odd

THEOREM 4. Suppose the resource level in (4) has a positive carrying capacity (alo/a11>O). Then, the food chain as modelled by system (4) is persistent provided p > 0 : it is not persistent/f#<0.

Proof First, we review some appropriate properties o f the dynamical system (4). Bounding hyperplanes o f R +" of the form H j = { ( x l , x 2 , x3 .... ,xn) R~_ :xj=0} are invariant manifolds for (4). The intersection of the co-limit set f~4, o f a solution q~ (which is bounded by T h e o r e m 3) with such a hyperplane is therefore a compact invariant set which contains a minimal compact invariant set (Nemytskii and Stepanov, 1960). A classical result o f Birkhoffimplies that every trajectory in this minimal set is recurrent. When extinction o f some constituent o f the food chain represented by (4) occurs, its model equivalent is a trajectory approaching a bounding hyperplane. This extinction trajectory converges as t ~ o o (since, as previously remarked, finite extinction time is not possible) to a set containing recurrent solutions.

884

T H O M A S C. G A R D A N D T H O M A S G. H A L L A M

Suppose now that # > 0 and, for the purpose o f contradiction, that system (4) has a solution qS=(~b 1, q52,..., ¢,) with qS(0)eR~ which satisfies limt~o~qSj(t)=0, for some index j. First, we note that limt.ooqSi(t)=0, j < i < n. This follows since,

(t);C-aj+ q-aj+1,j~j(t)-aj+1,j+2¢j+2(t)~ (E

i,o

if

j÷l
aj + 1,0 JCaj + 1,jOj(t)], if j + 1 -- n

J"

So, for sufficiently large t, qSj+ ,(t) < e j+ ,(t)[ - aj+ a,o/2]. Thus limt_.o~¢j+l(t)=0 , and so continuing this argument, if necessary, we prove the assertion. Hence, if qSj(t)~0 for some j as t ~ o o , .then 4~,,(t)~0 as t--, oo. Thus the assumption that ej(t)--,0 as t--, 0% for some j, implies that ~¢ ~ H,. We shall now show that ~b cannot have its assumed asymptotic behavior. This is accomplished by showing that for some r i>O, i= 1, 2,..., n, and for SOlIle )., h

p(x)= [[ x? i=1

is a persistence function on the hyper-rectangular slab

S = {(xl,x2 .... ,x,)e R"+ :0
/~ = [rl(alo -- allq51 --a12~b2) + ~ rj(--ajo -]-aj,j_ lq~j_ 1 - - a j , j + l(/)j+ 1) j=2

+r,(-a,o+a,,,-14), =p

I rlal°-j~-2rjajo + (r2a21 - rl al 1)41 +j =Z3

1)] (5)

1-",,

PERSISTENCE IN FOOD WEBS--I. LOTKA VOLTERRA FOOD CHAINS

885

Now, choose the rj's, 1 < j < n, so that ri/r I = all/a21 and rj/rj 2 - a j - 2 , j - 1 / a/j , j - l ,

j=3,4,...,n.

Using these relationships to determine the quotient rj/r I as a function o f the coefficients, we find i f j is even and 4 < j < n, ~ a2i 2,2i- 1. 11 a 2 1 i = 2 a2i,2i-1

rj a l l 1"1

(6a)

while ifj is odd and 3 < j < n, Fj

=

( j - 1)/2a2 i

H

t'1

i=1

- 1,2i a2i+1,2i

(6b)

Substitution o f (6) into (5) leads to f i = p [ r l p - - r , _ la,_ l,n~,].

(7)

Now by choosing 2 sufficiently small, we have, since # is assumed positive, that the expression in the square brackets in (7) is positive on S. So on S fi > 0 and as previously remarked, extinction cannot occur. This contradiction establishes that # > 0 is sufficient for the persistence of (4). To demonstrate that # < 0 implies extinction, the extinction function x, , /=1

where all o f the ri's are chosen according to the scheme (6) used above, is employed. Then, from (4), it follows that e satisfies the inequality i=e[rl#-

r,_ lan_ 1 ,nXn'] ~ rl#~'.,

with a comparison equation o f extinction type. Theorem 2 implies that the food chain is not persistent. We now consider a food chain modelled by (4) in which the resource level grows exponentially; that is, a11=0. In this setting, the persistence parameter #0 depends more intricately on the dimension of. the system than the previous case with positive carrying capacity. If. the food chain length n is either 2m + 1 or 2m + 2 for some positive integer m then #0 = a l o

-

j=l

a2j+l,O

~

a2i-l'2i

i = l a2i+1,2i

886

T H O M A S C. GARD AND T H O M A S G. HALLAM

THEOREM 5. Let a ~ = 0 . The food chain modelled by system (4) is persistent provided ~o > O; it is not persistent if #o < O.

Proof Theorem 3, which established the boundedness o f solutions o f (4) assuming a ~ > 0 , was required in the p r o o f o f Theorem 4. The analogous lemma employed here is that any solution ~b o f (4) with ~b(0)eR~ such that l i m t ~ q ~ i ( t ) = 0 for some j, is bounded. To establish this, we first note that lim~_~q~j(t)=0 implies l i m t ~ b i ( t ) = 0 for i > 1, as shown in Theorem 4. It suffices then to prove the assertion in the case that ~b is such that l i m ~ b , ( t ) = 0 . To this end, we use the classical Lotka-Volterra auxiliary function

V(XI,X2,-.., Xn)= ~, O~i(Xi--gi- f i / l o g

xi/gi)

,

i=1

where c~, ,qi are positive parameters to be chosen later. Boundedness of ~b will follow if it is shown that v(t)=v(O(t)) is eventually nonincreasing. Employing (4) with some rearrangement leads to n-1 ~(~bl, (])2 . . . . , (~n) =

E

[ - - °~iai, i +1 ~- O~i+

lai+ 1,i](Oi

--

fl,)(qSi+ 1 -

fli+l )

i-1

+ cq (q~l -- gl)(a, o - a, 2fi2) n--1 i=2

+o~,(~,--fl,)(--a,o +an,n- lfln- a). The parameters ~, i = 1,..., n are selected so that the first sum of n - 1 terms vanishes. I f n is even, it is possible to choose the g~ so that each of the remaining terms is zero:; thus, in this case, ~, g~ exist so that /5= 0 from which the boundedness o f q~ follows. Whenever n is odd, we can choose the g~ parameters in such a way that all terms vanish with the exception of the last one; hence, =

-/L)(-

a°o + a°,._ m

=

17 a2i+

i= l a2i-

This leads to m i= 1 ~ 2 i

1,2i

.2,

l,2i

PERSISTENCE IN FOOD WEBS--I. LOTKA VOLTERRA FOOD CHAINS

887

which, since #0 > 0, satisfies ~(t)< 0 for all t sufficiently large. Therefore, any solution which does not persist is bounded. The remainder of the p r o o f is the same as that of Theorem 4 up through the calculation o f t i ; from (5) with all = 0 we obtain

15>=Pl r1(al o --i~=2 r~11aio) -~- i ~= 3 ( r i a i ' i - l - - r i - 2 a i - 2 " i - 1 ) ~ i - l - - r n - l a n - l ' " ~ ) n l "

N o w let n be written as n = 2 m + l = 1, 2,..., m as in (6b): rj

or n = 2 m + 2 .

(J~l)/2a2i "= 1

(8)

We choose r2j+l, j

1,2i '

1,2i

Since #o > 0, it follows that & alo--

Select the remaining

r2i + 1

).. a 2 j + l , o j=l

r2j , j= 1,..., m alo

--

~

>0. rl

(or m + t if n = 2 m + 2 )

(rJrl)a;o

so tha.t

> 0.

(9)

j=2

If q is the largest even index, q < n, we n o w adjust r2, r4,..., rq sequentially (to b o u n d successive terms on the right side o f (8) while preserving inequality (9)) according to the following scheme. Choose rq 2 possibly smaller than the original choice so that rq//rq- 2 >=aq _ 2.q- 1/aq, q - 1,.

In a similar manner, choose rq 4 smaller if necessary so that rq _ 2/rq _ 4 ~ aq - 4 , q - 3/aq - 2,q - 3.

Continue adjusting the r2i until r2 is revised. We n o w have

t5 ;>P[ri#o - rn

1Cln- 1,n4n]"

Proceeding as in Theorem 4, we can select 2 so that the expression in the brackets remains non-negative on S. This leads to the desired conclusion that #o > 0 implies system persistence.

888

THOMAS C. GARD AND THOMAS G. HALLAM

The p r o o f that #o < 0 implies nonpersistence is analogous to the similar portion o f the proof of Theorem 4. As an extinction function use m+ 1

~(xl, x 2 , . . . , x , ) = [ I

x~.-~

i=1

where r i are determined by (6b). Then, when n is odd, the differential equation /=rl~oe results; if n is even, we obtain

g=e[rl#o-r.-la. 1,.~)n], from which nonpersistence follows.

4. Comment and Comparison. While the above results are independent of system equilibrium considerations, the conclusions have a positive correlation with the location o f the equilibrium points when they exist. For convenience in demonstrating these and other relationships, we restrict consideration to food chains of length four with nonzero resource carrying capacity: x', = x l (al o - a11x~ - a~ 2 x 2 ) X~ = X2( -- a20 -f- a 21x1 -- a 2 3 x 3 )

(10) X; =x3(--a30

+a32x 2-a34X4)

X~ = X4( -- a40 if- a43x3).

The equilibrium points of (10) are indicated in Table I. I f we designate the ith coordinate o f equilibrium point I by xl, II by x~~, etc., the persistence Theorem 4 may be phrased as (10) is persistent if the coordinate equilibria satisfy the inequalities x v < x~v (with the sequential ordering :~1" v ~-xl-Iv ->xl" ni ~.._x~l > 0); or equivalently, x v > x TM (with the sequential ordering x v > x~v < x n' > 0); or equivalently v_ "Y3 <'X3~v

t 3 < x ' V~>

"''-

PERSISTENCE IN F O O D WEBS

I. LOTKA VOLTERRA F O O D CHAINS

889

TABLE I Trophic Level Coordinates and Influence Factor 1 ~ ~11 a12

I:

~ 2 -'a12a21 alia23

- * 3 ~ alla23a32 a12a21a34

~4

X1

X2

X3

X~

0

0

0

0

al°

0

0

0

£111 V <=>

A

a2~°

al°

alia2°

0

0

a21

[112

a12a21

A

II:

IIl:

A

IV:

<~

al° all

A

a2° t a~°a21 a23 alia23

--a3°

a32

a l 2~130

a l 2a21 a 3 0

al ta23a32

<=>

A

a2°

a~°

a21

a12

4

~

al la32 V

V;

V

<=>

V

~

0 A

alia2°

a23a4o

a12a21 al la23a4o

a4o

-- a30

atoa32

alla2oa32

a 2 1 a43

a 12a21 a 4 3

a43

a34

a12a34

a12a21a34

alla23a4oa32

a12a21a43a34

Equilibrium points o f (10) are listed hierarchically in terms of coordinates. " ~ " represents implications for the appropriate coordinate orderings. Trophic level influence factors for each level are listed (see text for explanation).

or equivalently,

xV>O. The last inequality is just a restatement of the persistence inequality # > 0 . Analogously, if the coordinate ordering is reversed, nonpersistence results. This equilibrium coordinate ordering indicates that introduction of a new top predator into an ecosystem would result in an order interchange of equilibrium populations on an alternating scheme. For example, in an aquatic system with trophic levels composed of phytoplankton, herbivores, and primary carnivores, the model conclusion predicts that the addition of a secondary carnivore reduces the equilibrium levels of the phytoplankton and primary carnivores while increasing the herbivore population.

890

THOMAS C. GARD AND T H O M A S G. HALLAM

The component ordering also indicates that the persistence extinction mechanism is mathematically one o f bifurcation associated with the parameter #; essentially the equilibrium V for (10) "splits" off from IV as the parameters transect from the extinction (/~<0) to persistence range (g >0). In the case o f a persistent system, the resource transfer is modulated by trophic level influence factors which determine to some extent the magnitude o f persistence. These factors are indigenous to the specific trophic level and relate persistence in the food chain hierarchy. If the model is interpreted as nutrient flow, energy flow, etc., componentwise throughout the system, trophic level 1 contribution factor, all/a12, is the persistence multiplicative effect o f level 1 on trophic level 2. Trophic level 2 has a multiplicative contribution factor o f (alzazl)//(alla23) o n trophic level 3 while level 3 has multiplicative contribution factor (allaz3a32)/(aizazla34) for level 4. The antipodal character of certain of the trophic level influence factors might not have been anticipated. In general, the bifurcation value is persistence indeterminant. In the noncarrying capacity model, Freedman and Waltman (1977) have shown, for the three dimensional system, # = 0 implies persistence. On the other hand, for the four dimensional food chain (10) with a l a = 0 there is a continuum of critical points in the positive XaX2X 3 space when # o = 0 . A subset of these satisfies x3 0 is equivalent to the existence o f a positive cone equilibrium. It is worth noting that for odd dimensional models (4) without carrying capacity, persistence (#o>0) results even though there is no-positive equilibrium.

PERSISTENCE IN FOOD WEBS I. LOTKA VOLTERRA FOOD CHAINS

891

5. S u m m a r y . T h e p e r s i s t e n c e - e x t i n c t i o n m e c h a n i s m for simple f o o d chains m o d e l l e d b y L o t k a - V o l t e r r a d y n a m i c s is d e t e r m i n e d b y a single p a r a m e t e r which d e p e n d s u p o n the m o d e l p a r a m e t e r s a n d the length of the f o o d chain. It s h o u l d be n o t e d t h a t this p a r t i c u l a r m o d e l f o r m u l a t i o n ignores m a n y i m p o r t a n t biological a n d p h y s i c a l features. H e n c e the conclusions should be studied as biological h y p o t h e s e s requiring corroboration.

T h o m a s G. H a l l a m gratefully a c k n o w l e d g e s Office of N a v a l R e s e a r c h .

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LITERATURE Coste, J., J. Peyrand and P. Coullet. 1978. "Does Complexity Favor Existence of Persistent Ecosystems?" J. Theor. Biol., 73, 359-362. Freedman, H. I. and P. Waltman. 1977. "Mathematical Analysis of Some Three Species Food-Chain Models." Math. Biosci., 33, 257-276. Gob, B. S. 1977. "Global Stability in Many-Species Systems." Am. Nat., 111, 135 143. Goodman, D. 1975. "The Theory of Diversity-Stability Relationships in Ecology." Q. Rev. Biol., 50, 237~266. Harrison, G. W. 1979. "Global Stability of Food Chains." Am. Nat., in press. Holden, C. 1977. "Endangered Species: Review of Law Triggered by Tellico Impasse." Science, 196, 1426 1428. Holling, C. S. 1973. "Resilience and Stability of Ecological Systems." Ann. Rev. Econ. Syst., 4, 1-23. LaSalle, J. P. 1976. The Stability of Dynamical Systems. Philadelphia: SIAM. May, R. M. 1974. Stability and Complexity in Model Ecosystems. 2nd Ed. Princeton: Princeton University Press. Maynard Smith, J. 1974. Models in Ecology. Cambridge: Cambridge University Press. Nemytskii, V. V. and V. V. Stepanov. 1960. Qualitative Theory of Differential Equations. Princeton: Princeton University Press. Paine, R. T. 1966. "Food Web Complexity and Species Diversity." Am. Nat., 100, 65-75. Patten, B. C. 1974. "The Zero State and Ecosystem Stability." Proc. 1st Int. Cong. Ecol., 'The Netherlands: The Hague. Rescigno, A. and K. G. Jones. 1972. "The Struggle for Life: III. A Predator-Prey Chain." Bull. Math. Biophys., 34, 521 532. Saunders, P. T. and M. J. Bazin. 1975. "On Stability of Food Chains." J. Theor. Biol., 52, 121-142. Yorke, J. A. and W. N. Anderson, Jr. 1973. "Predator Prey Patterns." Proc. Natn. Acad. Sci., U.S.A., 70, 2069 2071. RECEIVED 5-1-78 REVISED 10-25-78